New Perspectives in Nonlinear Partial Differential Equations Modeling Collective Quantum Dynamics

该项目致力于分析集体量子动力学的数学描述中产生的非线性偏微分方程，并应用于，例如，玻色-爱因斯坦凝聚。所考虑的数学方程大多是色散的，因此是保守的，但也将研究几个非保守（或耗散）的扩展。通过这样做，非线性薛定谔型方程和复杂的金斯堡-朗道型模型之间的联系将变得明显。在本项目中，PI将：（1）研究实验上用于产生量子涡旋的快速旋转势对有限时间爆破可能性的影响，并与最近发现的奇异环型解建立联系;（2）通过绝热微扰理论给出涡旋产生的时间依赖性描述;（3）建立非线性薛定谔方程与其它色散模型之间的严格联系：（4）探讨所考虑方程色散爆破的可能性;（5）分析了带有非线性阻尼项和其他耗散正则化项的Schr dinger型方程解的长时间行为。要部署的技术范围包括从正式分析方法的数值模拟到基于以下内容的完全严格的数学研究，例如，维里估计，多尺度技术，李雅普诺夫型泛函和浓度紧致方法。该项目旨在寻找多体量子力学中某些集体现象的严格数学描述。量子力学是现代物理学的两个基本组成部分之一，它涉及研究尺寸接近甚至小于原子尺度的力学系统。由E.薛定谔、狄拉克和冯·诺依曼在很大程度上是基于复值波函数的偏微分方程，它以概率解释描述粒子的状态。这些方程的解可以被看作是理论发展早期阶段的主要驱动力之一。本项目是在这些方程的数学分析和它们在理论物理中的应用之间的交叉点。这将有助于获得更深入的数学理解的定性行为的不同类别的非线性方程及其相互关系。对所提出的问题的回答将在广泛的数学科学中产生强大的影响，并可以被视为对物理学文献中大量非严格结果的补充。大多数拟议的研究将通过与数学分析，理论物理和数值模拟方面的同事进行国家和国际合作进行。

This project is devoted to the analysis of nonlinear partial differential equations arising in the mathematical description of collective quantum dynamics, with applications to, e.g., Bose-Einstein condensates. The mathematical equations under consideration will mostly be dispersive, and hence conservative, but several non-conservative (or dissipative) extensions will also be studied. By doing so, connections between equations of nonlinear Schrödinger type and complex Ginzburg-Landau type models will become apparent. In the present project the PI will: (1) study the influence of fast rotating potentials, experimentally used to create quantum vortices, on the possibility of finite time blow-up, and establish connections to recently found singular ring type solutions; (2) give a time-dependent description of vortex creation by means of adiabatic perturbation theory; (3) establish rigorous connections between nonlinear Schrödinger equations and other dispersive models; (4) explore the possibility of dispersive blow-up for the considered equations; (5) analyze the long-time behavior of solutions to Schrödinger type equations with additional nonlinear damping terms and other dissipative regularizations. The techniques to be deployed will range from numerical simulations over formal analytical methods to fully rigorous mathematical investigations based on, e.g., virial estimates, multi-scale techniques, Lyapunov-type functionals, and concentration compactness methods.This project aims to find a rigorous mathematical description of certain collective phenomena in many-body quantum mechanics. Quantum mechanics is one of the two fundamental building blocks in modern physics and it concerns the study of mechanical systems whose dimensions are close to, or even smaller than, the atomic scale. The mathematical formulation pioneered by E. Schrödinger, P. Dirac, and J. von Neumann is to a large extent based on partial differential equations for complex-valued wave functions, which describe the state of the particles in a probabilistic interpretation. It is the solution of these equations which can be seen as one of the main driving forces in the early stages of the development of the theory. The present project lies at the intersection between the mathematical analysis of such equations and their application in theoretical physics. It will help gain a deeper mathematical understanding of the qualitative behavior of the different classes of nonlinear equations involved and their relations with each other. Answers to the questions raised will have a strong impact as proven statements in a wide range of the mathematical sciences, and can be seen as complimentary to the vast amount of non-rigorous results available in the physics literature. Most of the proposed research will be conducted through national and international collaborations with colleagues working in mathematical analysis, theoretical physics, and numerical simulations.